NS: chapter 5 (nearly all of it, see web book)
Additional to text: Dominance, rationalizability
Experimental evidence (and evidence from the real world); supplements
… Problem set coming, heavy coverage of game theory!
Module context
Basic tools, the economic approach
The simple classical model, welfare results
\(\rightarrow\) and now Strategic uncertainty \(\rightarrow\) Game Theory
Is game theory useful???
Some people seem to think so:
Today:
Please have Turningpoint/Responseware available over next lectures
Previously: Each individual (consumer, firm… ) takes others’ choices as given
Now: Consider strategic interaction
My best choice may depend on your choice, & vice versa
One minute exercise
Find a situation—in business, government, fiction, history or your own life—
… where one party’s optimal choice depends on what another party does.
Write it down, give 1-sentence explanation of why it involves ‘strategic dependence’
Some possible examples:
Amount to bid at a first-price auction?
Whether Robotic Chef opens a new branch in Exeter, and where?
How hard to work towards a promotion at your job?
##Some examples
Is it better to get lunch at Comida or Pret?
What if your friends are going to Comida?
What if everyone and her cousin are going to Comida, so the queue is miles long?
What should Tim Cook charge for his new Iphone?
Does it depend on whether Samsung and LG…
… Sell their phones for £200, or £1000, or go out of business?
A language & framework for analyzing strategic situations
Solution concepts make predictions under given assumptions
Equilibrium defined as a baseline
Players
Strategies
Payoffs
Information
the decision makers in the game
\(2, 3, . . . , N\) players
Who are the players in the game ‘Chicken?’
Simple games: same as actions
Each player’s utility from the combination of each player’s strategies (and chance) in the game
May include ‘money earned’ + other considerations; all summarised in the payoff numbers
Each player’s goal: maximise her payoff (not just to ‘win’)
N Straight, S pulls off \(\rightarrow\) N ‘wins’, S ‘loses’
N pulls off, S straight \(\rightarrow\) N ‘loses’, S ‘wins’
To convey this game payoffs must follow: Win \(\succ\) tie \(\succ\) lose \(\succ\) crash
Example of payoffs in Chicken (as matrix)
what each player knows, at a particular point in the game, about payoffs and previous actions
For sequential games, players may or may not know other players’ previous actions
Consider… for the game you imagined earlier…
Who are the players
What are the actions/strategies
What are the payoffs?
Is it simultaneous or sequential?
Vote: A - confess, B - Silent
Vote: A - confess, B - Silent
Q1: What would you do?
Q2: What do you think game theory predicts people will do?
Q4: Which outcome is definitely NOT Pareto-optimal (for the prisoners)?
A. Both players confess
B. Both players are silent
C. A confesses and B is silent
D. B confesses and A is silent
E. All outcomes may be Pareto-optimal
A Prisoner’s Dilemma be like
Two Players: (A and B, row and column, whatever)
Strategies (Actions): ‘Cooperate’ with other prisoner (C) or defect (D) and confess
In normal form:
In normal form:
To be a prisoner’s dilemma, payoffs must satisfy \(T > R > P > S\)
Common knowledge
Why is common knowledge important?
You may be stuck on an island with 100 blue-eyed people and 900 brown-eyed people
(Links to video and cartoon versions in handout)
Island with 1000 people. 100 w/ blue eyes, 900 brown eyes. No reflective surfaces.
By strict custom:
American tourist says ‘It’s so nice to see one or more people with blue eyes in this part of the world’.
Q: What effect, if any, does this faux pas have on the island?
GT predicts: Statement made common knowledge \(\rightarrow\) all 100 blue-eyed suicides 100 days later
a strategy for player A that gives him the highest payoff of all his possible strategies, given that the other player(s) play S
Simple prediction: a ‘rational’ player will play a dominant strategy if she has one.
A simple prediction: a rational player will never play a dominated strategy
Prediction of ‘players play dominant strategies’
…in Prisoner’s dilemma?
But in other cases, this may have no clear prediction
Extending this …
Rationality assumption: the players are rational
Common Knowledge of Rationality assumption:
Players know all other players are rational. They know all players know all players are rational. They know… (all players know … ad infinitum) … all are rational.
Thus they know other players will never play a dominated strategy, and eliminate these from consideration.
Thus they won’t play a strategy if another strategy is always better against this reduced set of possibilities.
…Etc.
… the process of ‘Iterated Strict Dominance’ (ISD). \(\rightarrow\) Strategies that survive ISD: ‘rationalizable’
ISD example; may yield a unique prediction
But there may be no dominated strategies, or ISD may leave many possibilities
How economists consider strategic interaction
Basic notation & grammar of game theory
Key concept: Common Knowledge
Predictions: Dominant/dominated strategies, ISD/Rationalisability
…
Repeated games: definite time horizon; indefinite/infinite
Repeated games: indefinite/infinite
Continuous Action games
Experimental evidence: What is a laboratory experiment in Economics?
Market equilibrium (recapping): given the equilibrium price & quantity, no market participant has an incentive to change her behaviour.
Similar concept for strategic settings:
If I play my BR to your chosen strategy and you’re playing your BR to mine, neither of us has an incentive to deviate — an equilibrium.
All games have at least one Nash equilibrium
Caveat: we might not expect such play to actually occur (particularly not in one-shot games)
First method: Inspection
Check each outcome.
Either player has incentive to unilaterally deviate?
If not \(\rightarrow\) it’s (outcome of) a NE.
Second method to find NE – Underlining
Underline payoffs for BR’s of each player.
Outcome with 2 underlines \(\rightarrow\) outcome of a NE (strat. profile).
Find equilibrium via each method:
If eliminating dominated strategies yields a single prediction for each player, these form a NE (profile).
Same for ISD (rationalizability) … if it leads to a unique prediction, it’s a NE.
HOWEVER: not every NE involves dominant strategies
In-class experiment: BOS & coordination; need 2 volunteers
2019-20: know the basic principles; you don’t need to compute mixed strategies
Remember: there is always at least one NE. If there is no pure strategy NE, there will be a NE in mixed strategies.
Intuition
Want to derive the best response functions, find intersections.
Let \(h\): probability husband chooses Ballet, \(w\): probability wife chooses Ballet
Rem: Wife chooses Ballet (Box) \(\rightarrow\) Husband’s BR is Ballet (Box)
Using ‘EU’: Compute Husb’s payoff from playing Ballet & from Boxing if Wife chooses Ballet w/ prob. w
\(\rightarrow\) ‘Threshold’ prob. \(\bar{w}\) makes him indifferent; for \(w<\bar{w}\) , he chooses Box; for \(w >\bar{w}\) he chooses Ballet.
… similarly, wife indifferent btwn her 2 choices for some prob. \(\bar{h}\); for \(h<\bar{h}\), she chooses Box, for \(h>\bar{h}\) she chooses Ballet.
If she thinks he goes below 1/3 of the time she goes Boxing
If she thinks he goes exactly 1/3 of the time she is indifferent
Three equilibria: both Boxing, both Ballet, mixed strategy; are any more reasonable as predicted outcomes?
Also see further discussion in handout about whether and why a mixed strategy equilibrium is a reasonable prediction, and what the intuition is.
There’s always one or more they say but sometimes it is mixed
you find one where the best responses meet at some point fixed
you only know you won’t regret the strategies it ‘picksed’
I hear that it’s prediction power is shabby
It might not yield the best outcome and might not yield the worst
In prisoner’s dillemas a confession is coerced,
I hate to have to say it, but I very firmly feel Nash Equilibrium’s not an asset to the abbey
I have to say a word on its behalf: multiple-equilibria make me laugh
How do you solve a problem like multiple-equilibria?
Many a thing you’d like old Nash to tell you
Many a thing you thought you’d understand
But how do you find a way to predict equilibrium play?
How do you keep a wave upon the sand?
Oh, how do you solve a problem like multiple-equilibria?
How do you hold a moonbeam in your hand?
‘The Big Game’
A previous year’s results (2016, with slightly different payoffs)
| Share chose | .. squared | Pay if match | E(Pay) | |
|---|---|---|---|---|
| 1 | 0.26 | .068 | 2 | 0.52 |
| 2 | 0.21 | .044 | 2 | .42 |
| 3 | 0.11 | .012 | 1 | .11 |
| 4 | 0.16 | .026 | 2 | .32 |
| 5 | 0.26 | .068 | 2 | .52 |
| Wtd avg | 0.22 | 0.05 | 0.42 |
In this trial the ‘middle square’ yielded payoffs of 1
We refer to refinement criteria and focal points
Equilibrium with the highest payoffs for both?
In BOS this rules out mixing (payoffs 2/3, 2/3 for h,w respectively)
But doesn’t say whether it’s Box, Box (payoffs 2,1) or Ballet, Ballet (payoffs 1,2)
Choose the ‘symmetric equilibrium?’ … here, mixing
Choose the one that seems like a ‘focal point’? (rem, the ‘big game’)
A year after graduating you come back for Alumni Weekend. You are supposed to meet the veterans of this module for a night of festivities but can’t remember where or when.
The internet does not exist.
Where do you go? Write it down on a piece of paper
What if you are meeting for a reunion in London, and no one has internet or phone access?
Where do you go?
Write it down
Refinements are real-
(ly) important to some academic referees:
However, I am dismayed that you chose to ignore my suggestion to come up ”with a reasonable refinement that kills all C-F equilibria." I agree that introducing additional types and rounds may smooth out the extreme outcome in the C-F equilibria. But, it is perfectly fair to evaluate the predictions of your actual model and it does not include these features.
Consider Battle of Sexes, but now Wife chooses first, Husband observes this and then chooses. What do you think will happen? Vote:
A. Wife: Ballet, Husband: Box
B. Wife: Ballet, Husband: Ballet
C. Wife: Box, Husband: Ballet
D. Wife: Box, Husband: Box
I.e., each ‘game’ starting from a point where a player knows where she is (knows previous choices)
You can solve for this with ‘backwards induction’ (BWI)
Example: BWI for BOS
Formally specify (SP)NE strategies for above game:
a NE, not SP: {Wife: Boxing; Husband: Boxing, Boxing}
SPNE: {Wife: Ballet; Husband: Ballet, Boxing}
(reading Husband’s decision nodes left to right; please specify this)
It can get fancier
In ‘normal’ (matrix form), stating complete contingent strategies:
Find the SPNE; ‘state the complete contingent strategies’
A: S1, N3, N5
B: s2, n4, n6
A ‘stage game’ is a simple (matrix) game that may be played repeatedly
Note: In repeated (as in sequential) games, a strategy is a ‘complete contingent plan’; specifies what the player will choose at every decision node
Whenever the stage-game is repeated, repeated play of the stage-game equilibrium is an equilibrium of this repeated game
Is there a way in which we can sustain cooperation in a finitely repeated Prisoners’ Dilemma?
No.
Suppose we repeat the Prisoners’ Dilemma a finite (T) number of times, e.g., 10 times.
What is the subgame perfect equilibrium?
Backwards induction:
Thus each Confesses in every period \(\rightarrow\) Can’t sustain cooperation.
Game played repeatedly for potentially infinite number of periods
Indefinitely repeated Prisoners’ Dilemma
Consider the following trigger strategy (for both players):
Is this a SPNE?
The payoff from staying Silent (cooperating) each period is:
\[-2 \times (1 + g + g^2 + g^3 + . . . )\]
The payoff from Confessing right away (after which both players Confess always) is:
\[ -1 + -3 \times (g + g^2 + g^3 + . . . ) \]
Formula for a geometric series (where \(0<g<1\)): \[g + g^2 + g^3 + g^4 ... = g/(1-g)\]
Thus cooperation in a single period is ‘weakly preferred’ (at least as good) if
\[(-2) \times (1 + g + g^2 + g^3 + . . . ) \geq (-1) + -3 \times (g + g^2 + g^3 + . . .)\]
\[g + g^2 + g^3 + . . . \geq 1\]
\[\frac{g}{1-g} \geq 1\]
\[g \geq \frac{1}{2}\]
\(\rightarrow\) cooperation can be sustained if the ‘probability of play continuing’ (or ‘patience’) is high enough; here, above 1/2.
GT for cases where people choose ‘continuous’ actions, e.g.:
With continuous actions…
\(\rightarrow\) Payoff functions, continuous ‘best response functions’; rather than a payoff matrix
Continuous games with differentiable payoffs, finding NE:
Write down payoff functions. (Differentiable?)
Max each player’s payoff wrt own strategy; other player’s payoff as a ‘parameter’ \(\rightarrow\) Players’ best-response functions.
Look for intersection(s) \(\rightarrow\) Nash equilibrium/equilibria
Shepherds A & B graze \(s_A\) & \(s_B\) sheep, respectively, in same meadow
More sheep graze \(\rightarrow\) less grass for each \(\rightarrow\) less wool/milk/meat per sheep
Benefit per sheep raised: \(120-s_A-s_B\)
Total benefit from raising \(s_A\) and \(s_B\) sheep:
Marginal benefit of ‘me adding’ another sheep:
For simplicity, assume zero marginal cost
Set MB=MC (here MC=0)
Rearrange above to solve for BRFs:
Solve for Nash equilibrium:
\[s_A = 60-\frac{1}{2} s_B\]
Plug this value of \(s_A\) into the BRF for B, solve for \(s_B\), then for \(s_A\)
\[s_B = 60-\frac{1}{2} s_A = 60 - \frac{1}{2}(60-\frac{1}{2} s_B)\]
\(s_A* = s_B* = 40\)
Shepherd A: \(s_A = 60-\frac{1}{2} s_B\)
Shepherd B: \(s_B = 60-\frac{1}{2} s_A\)
\(\rightarrow s_A* = s_B* = 40\)
Why the ‘Tragedy’ of the Commons?
Each grazes until his private marginal benefit equals his cost (0):
\(120-2s_A-s_B = 0\)
Does not consider the harm he imposes on the other guy
Above solves to \(s_A^{*} = s_B^{*} = 40\).
Each get payoff \(40 \times (120 - 40 - 40) = 40 \times 40 = 1600\)
To max total payoffs, set \(S=s_A+s_B\) to max
\[S\times(120-S)\]
Concave problem, set first-derivative to zero: \[120 - 2S = 0 \rightarrow S^*=60\]
Leading to total payoff \(60 \times (120 - 60) = 3600\). If split evenly (each graze 30), get 1800 each.
Sadly, this problem (literally!) is still relevant
E.g., FEELE lab at Exeter
Cooper et al (1996)
Cooperation even in anonymous ‘1-shot’ games (different opponents each time)
Cooperation declines somewhat over time, but not to zero
Mix of other-regarding and selfish types
Proposer goes first, proposes split of the ‘pie’, anything between 0%-100% and 100%-0% inclusive.
Responder can accept, or reject and get nothing
What happens in experiments?
Potential explanations
PD: Does money measure the true payoffs? Do we have the ‘control’ to test the model?
Do they resemble the ‘real world’ group of interest? (e.g., firms, countries, taxpayers, voters, home-buyers)
Right preferences and experience?
Are the (small) stakes relevant?
Are the right ‘environmental characteristics’ present?
Does the ‘imposed model’ apply?
Observed self-conscious environment, perhaps made aware of contrasts